The hypersurfaces with conformal normal Gauss map in H^n+1 and S_{1}^{n+1}

Abstract

In this paper we introduce the fourth fundamental form for the hypersurfaces in $H^{n+1}$ and the space-like hypersurfaces in $S_1^{n+1}$ and discuss the conformality of the normal Gauss maps of the hypersurfaces in $H^{n+1}$ and $S_1^{n+1}$. Particularly, we discuss the surfaces with conformal normal Gauss maps in $H^3$ and $S_1^3$ and prove a duality property. We give the Weierstrass representation formula for the space-like surfaces in $S_1^3$ with conformal normal Gauss maps. We also state the similar results for the time-like surfaces in $S_1^3.$

18 pages